Question

If function $f\left(x\right)$ is continuous in interval $\left[-2,2\right]$, find the value of $\left(a+b\right)$ where
$f\left(x\right)=\frac{\sin ax}{x}-2,$ for$-2\le x<0$
$=2x+1,$ for$0\le x\le 1$
$=2b\sqrt{x^2+3-1},$ for$1<x\le 2$

Answer 1

${lim}_{}h\to 0f(0-h)=\frac{\sin a(-h)}{-h}-2\Rightarrow \frac{\sin \left(ah\right)}{ah}\times a-2\Rightarrow a-2f\left(0\right)=2\times 0+1=1=a-2a=3f\left(1\right)=2\times 1+1=3f(1+h)=2b\sqrt{{(1+h)}^2+3-1}=2b\sqrt{3}=3b=\frac{\sqrt{3}}{2}a+b=3+\frac{\sqrt{3}}{2}$